E. Application Domains and Software
• Mixed integer programs can be used to formulate just about any discrete optimization problem. They are heavily used in practice for solving problems in transportation and
manufacturing: airline crew scheduling, vehicle routing, production planning, etc. Michael Tricks
Operations Research Page contains pointers to many web sites on operations
research in general, including math programming, specific methods, and specific problems or problem areas. The following are good journals to look at for technical papers applying
integer programming: Operations Research
, Mathematical Programming
Series A and B, and
SIAM Journal on Optimization .
• Commercial code includes IBMs Optimization Subroutine Library OSL
, CPLEX
, and XPRESS-MP by
Dash . The University of Karlsruhe has a list of
23 commerical solvers , with
some comparison information. Compass Modeling Solutions
, provides an AMPL ATT
Mathematical Programming Language interface to some commercial solvers. There are
other modeling languages, sometimes provided with the MIP package http:www.cs.sandia.govoptsurveymip.html
IV. Problem description A. Harvest areas
• A district is divided into harvest areas. Annual planning starts with a list of areas, identified as suitable for harvest during the next 1.5-2 years. Each harvest area is unique with its own
properties. It varies in size and in available volumes of assortments, and areas require between 1 and 20 days of work to harvest. The operations are either final felling or thinning.
• The production time depends on the average size of the trees and the equipment of the harvest team. Each area is also connected to a particular road or road group. Table 1 gives
typical data for five areas in the case study. The name of the areas, the number of standard hours to harvest the area, the average tree diameter, and quantities of the different
assortments are given.
• The accessibility profile code implies a certain annual profile of the accessibility, i.e., a certain grade of accessibility for each month. Areas with accessibility profile 5 correspond to
areas with soft ground, and areas with accessibility profile 1 correspond to areas with high accessibility, possible to harvest during thawing. There is generally a shortage of areas with
a good accessibility profile.
Table 1. Information used for five harvest areas.
B. Harvest teams
• There are a number of teams working full time in a district. A harvest team consists of a harvester, a forwarder, and two working groups with two people in each. The harvester fells,
Universitas Sumatera Utara
bucks, and piles the trees. The forwarder collects the log piles and moves them to pick-up points adjacent to the forest road. Some teams concentrate on final felling and others on
thinning. Each team has a unique capacity with respect to average tree diameter size and efficiency rate to estimate the specific number of standard hours needed for harvesting,
depending on their equipment. Each person in the teams travels back and forth to a home base each working day. Each combination of team and harvest area corresponds to a
particular cost.
C. Wood-processing facilities
• The demand for each mill saw-, pulp-, or paper-mills is given per month and assortment. Each mill has a given storage capacity that can be used for a fixed cost per cubic meter.
Some mills also have terminals for storage, which are outdoor locations where the logs are stored, adjacent to the mill. The transportation cost depends on assortment and distance
and is assumed proportional to the volume transported. An example of demand for the first 6 months for four mills is given in Table 2.
Table 2. An example of demand levels x1000 m
3
for four mills during a 6-month period.
D. Roads
• Individual forest roads are aggregated into a group of forest roads and assigned a specific ID number. Some areas are connected directly to the state road network. Most areas are
connected to a forest road group, which in turn is connected to the state road network. Road groups are used because single roads are regarded as too detailed. Also, there are areas
that are connected to forest roads, which are connected to another forest road group before the state road network is reached. This hierarchy decides how maintenance and snow
removal must be carried out.
• Accessibility of the state roads is guaranteed. During parts of the year, some roads must be avoided because of soft ground, and some forest roads need restoring to be accessible,
e.g., snow removal in wintertime. The road dependence is illustrated in Fig. 4. In this example, areas 1, 2, 3, and 4 are connected to road group 1, which is connected to the state
road network. Areas 5, 6, 7, and 11 are connected to road group 2, and areas 8, 9, 10, to road group 3. Road group 3 is connected to road groups 1 and 4, and road group 2 is
connected to road groups 1 and 5, before reaching the state road network.
Figure 4. Illustration of a possible road network.
Universitas Sumatera Utara
V. Mathematical Formulation
Kategori